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Understanding Ohm's Law is fundamental to grasping the concepts of both series and parallel circuits. Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and is given by the formula:

• \( I = \frac{V}{R} \)

Here, \( I \) is the current in amperes (A), \( V \) is the voltage in volts (V), and \( R \) is the resistance in ohms (\( \Omega \)).
In our example, when you connect a battery with a voltage of 12 volts to a circuit, you can determine how much current flows through resistors by using this relationship.
Ohm's Law is crucial as it helps us understand the behavior of electrical components when they are connected to a power source. By applying it to both series and parallel circuits, we can calculate the current and verify how resistances affect the circuit's operation.

In a series circuit, resistors are connected one after another, forming a single pathway for the electric current.
This means that the same current flows through each resistor.
The total or equivalent resistance \( R_{eq} \) of resistors in a series circuit is simply the sum of their individual resistances:

• \( R_{eq} = R_1 + R_2 + R_3 + ... \)

For our given problem, with three resistors each being 9.0 ohms, the equivalent resistance is calculated as \( 27.0 \Omega \).
This greater resistance means that the current flowing through the series circuit will be less, due to the greater opposition posed by all the resistors combined.
Using Ohm's Law, the current can be calculated as \( I = \frac{V}{R} = \frac{12V}{27.0\Omega} = 0.44A \).
Each resistor will have the same current of 0.44 Amps, which is a typical characteristic of series circuits.

A parallel circuit has multiple paths for the electricity to flow, as each resistor is connected across the same two points. In such arrangements:

• The voltage across each resistor is the same, equal to the battery's voltage.
• The current may vary through each path, depending on the resistance in that path.

To find the equivalent resistance \( R_{eq} \) in a parallel circuit, we use the formula:

• \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... \)

For the given resistors, this results in \( R_{eq} = 3.0 \Omega \).
This lower equivalent resistance increases the total current in the circuit.
Using Ohm's Law, you calculate the total current as \( I = \frac{V}{R} = \frac{12V}{3.0\Omega} = 4.0A \).
Each resistor has the same voltage of 12V, and the current through each resistor can be found using \( I = \frac{V}{R} = \frac{12V}{9.0\Omega} = 1.33A \).
Hence, in a parallel circuit, while the voltage remains the same across all resistors, the current is divided among the paths.